How do we measure openings in surfaces? Say, for example, you close up a draw-string bag. Do you think of the decreasing area or circumference of the hole? This distinction is essentially the relationship between magnetic field and the vector potential.
Consider an open drawstring bag; the magnetic flux through the opening is:
Since the hole has both a perimeter and an area, we could have represented the magnetic flux as a line integral around the opening, rather than as a surface integral through the hole. The vector potential is just this, the quantity which when integrated around an open surface, represents the magnetic flux through the surface. Thus:
To continue with the bag analogy, the open surface need not be flat. We can wrap the bag around anything and calculate the flux through the hole.
Imagine, hypothetically, that we instead wanted to calculate the electric, rather than magnetic, flux. We could choose to wrap the bag around some charge, or not, and we would get different answers either way. Thus, defining a vector quantity related to the electric flux would be meaningless.
Luckily we do not have that problem, since no absolute charge of northness nor southness exists. The magnetic flux through our bag only depends on what passes through the opening, and does not depend on the shape of the surface of choice. This is what makes the vector potential such a powerful tool.
Franz Ernst Neumann introduced an expression for the vector potential in 1845. Wilhelm Weber, soon, suggested a different expression in 1846. In 1847, William Thomson suggested an expression for the vector potential in a number of cases, including the vector potential of a small magnet in terms of the magnetic moment, which we use extensively in this chapter. 
The vector potential was controversial throughout the 1800s, because of its apparent fatal flaw of non-uniqueness. Recall that both Neumann and Weber published different expressions, which were both physically equivalent, as later shown by Hermann von Helmholtz . By the end of the nineteenth century, most physicists agreed with Oliver Heaviside and Heinrich Hertz, and thought that it added no additional physical significance.
Only a few decades later, however, Albert Einstein and others found that it fit beautifully within the theory of relativity because A and V together form a relativistically covariant four dimensional space-time vector from which the electric and magnetic fields can be derived.
This rapid change in fate of A coincided with a fall in the prestige of the vector H and the downfall of the pole model. Either Petrus Peregrinus was correct, or he was not. If magnetic monopoles exist, than A simply becomes a function of convenience. The opposite is true of H.
Yakir Aharonov and David Bohm, addressed this question in 1959, when Aharonov was Bohm’s doctoral student. They developed thought experiments which would yield results that depended on the vector potential, even when there were no fields present. We cannot say it any better than them:
In classical mechanics, we recall that potentials cannot have such significance because the equation of motion involves only the field quantities themselves. For this reason, the potentials have been regarded as purely mathematical auxiliaries, while only the field quantities were thought to have a direct physical meaning.
In quantum mechanics, the essential difference is that the equations of motion of a particle are replaced by the Schrodinger equation for a wave. This Schrodinger equation is obtained from a canonical formalism, which cannot be expressed in terms of the fields alone, but which also requires the potentials.
In this Chapter we introduce the magnetic vector potential, A, and use it to simplify the calculation of magnetic fields surrounding permanent magnets. In order to do this, however, we must define a new vector derivative, called the curl.
The corresponding fundamental theorem of calculus is called Stokes Theorem. It was developed by William Thomson and sent to George Gabriel Stokes in a letter. Stokes then put it as a question on the 1854 Smith’s Prize Exam for graduate students at the University of Cambridge, where James Clerk Maxwell sat for the exam. He won the prize having done particularly well on that question.
 See the review article: A.C.T. Wu and C.N. Yang, Evolution of the Concept of the Vector Potential in the Description of Fundamental Interactions International Journal of Modern Physics A 21 (2006) 3235-3277, World Scientific Publishing Company
 Y. Aharonov and D. Bohm, “Significance of Electromagnetic Potentials in the Quantum Theory,” Physical Review, 115 (1959), 485-491.